Handbook of c Higher Education Press Group Actions (Vol. IV) and International Press ALM41, Ch. 4, pp. 139{188 Beijing-Boston Mostow type rigidity theorems Marc Bourdon ∗ Abstract This is a survey on rigidity theorems that rely on the quasi-conformal geometry of boundaries of hyperbolic spaces. 2010 Mathematics Subject Classification: 20F65, 20F67, 20F69, 22E40, 30C65, 30L10. Keywords and Phrases: Hyperbolic groups and nonpositively curved groups, asymptotic properties of groups, discrete subgroups of Lie groups, quasiconformal mappings. 1 Introduction The Mostow celebrated rigidity theorem for rank-one symmetric spaces states that every isomorphism between fundamental groups of compact, negatively curved, lo- cally symmetric manifolds, of dimension at least 3, is induced by an isometry. In his proof, Mostow exploits two major ideas: group actions on boundaries and reg- ularity properties of quasi-conformal homeomorphisms. This set of ideas revealed itself very fruitful. It forms one of the bases of the theory of Gromov hyperbolic groups. It also serves as a motivation to develop quasi-conformal geometry on metric spaces. The present text attempts to provide a synthetic presentation of the rigidity theorems that rely on the quasi-conformal geometry of boundaries of hyperbolic spaces. Previous surveys on the subject include [74, 34, 13, 102, 80]. The orig- inality of this text lies more in its form. It has two objectives. The first one is to discuss and prove some classical results like Mostow's rigidity in rank one, Ferrand's solution of Lichn´erowicz's conjecture, the Sullivan-Tukia and the Pansu ∗Universit´eLille 1, D´epartement de math´ematiques, Bat. M2, 59655 Villeneuve d'Ascq, France. E-mail: [email protected]. Supported in part by ANR project \GdSous" and by Labex Cempi. 2 M. Bourdon quasi-isometry rigidity theorems. The second one is to present more briefly some of the numerous recent advances and results based on the quasi-conformal geometry of boundaries. The paper starts with three preliminary sections. Section 2 is a survey on Gro- mov hyperbolic spaces, their boundaries and CAT(−1)-spaces. Section 3 concerns rank-one non-compact symmetric spaces. We describe in detail their boundaries in relation to the nilpotent structure. Section 4 presents the geometric analytic tools that serve in the sequel. This includes the convergence property, the Sullivan char- acterization of M¨obiushomeomorphisms, the notion of Loewner spaces and the regularity properties of quasi-conformal homeomorphisms. The heart of the paper is formed by Sections 5, 6, 7, 8. Each of them consists of a main part with major statements and sketches of proofs, and of a succinct exposition of several related results. Section 5 is devoted to the Mostow original theorem and its proof. We also state some generalizations, including Besson-Courtois-Gallot's theorem. In Section 6, Ferrand's theorem and Sullivan-Tukia's theorem are stated and proved by using the zoom method. We also present Tukia's proof of Mostow's theorem (for real hyperbolic spaces). Section 7 focuses on rigidity of quasi-isometries. A detailed sketch of the proof of Pansu's theorem is given. Right-angled Fuchsian buildings are also discussed. Finally, Section 8 presents some recent developments and perspectives, including Cannon's conjecture and the combinatorial Loewner property. Clearly, several results presented in this paper would deserve a more detailed exposition. Moreover only very few aspects of rigidity are treated. In particu- lar, infinitesimal rigidity, higher rank rigidity, superrigidity, harmonic maps and bounded cohomology, do not appear at all. This is due to my own limitations. It is my pleasure to refer to P. Ha¨ıssinsky’ssurvey [78], in this volume, for a complementary viewpoint of some of the subjects discussed here. Acknowledgements Parts of the paper are issued from minicourses I taught. I am thankful to their organizers { Kingshook Biswas, Pralay Chatterjee, Swiatoslaw´ Gal, Mahan Mj, Damian Osajda. I acknowledge Peter Ha¨ıssinsky for the mini-course we gave together in Bedlewo. I thank the editors Lizhen Ji, Athanase Papadopoulos, Shing- Tung Yau, for the opportunuity of publishing this work in \Handbook of Group Actions". I am grateful to the anonymous referee and to Athanase Papadopoulos for their help in improving the exposition. Notation and convention In a metric space X, we denote by NR(Y ) the R-neighborhood of a subset Y ⊂ X, i.e. NR(Y ) := fx 2 Xj dist(x; Y ) ≤ Rg. The Hausdorff dimension of X is denoted by Hausdim(X). The homeomorphism and isometry groups of X are denoted by Homeo(X) and Isom(X) respectively. A map f : X ! Y between metric spaces Mostow type rigidity theorems 3 is a homothety if it is an isometry after possibly rescaling the metric of X or Y by a multiplicative positive constant. Two real-valued functions f; g defined on a space X are said to be comparable, and we write f g, if there exists a constant C > 0 such that C−1f ≤ g ≤ Cf. We write f . g if there is a constant C > 0 such that f ≤ Cg. 2 Gromov hyperbolic spaces Hyperbolic spaces and hyperbolic groups were defined by Gromov [73] in the mid- dle of the eighties. We give a brief survey on hyperbolic spaces and their bound- aries, see e.g. [73, 56, 72, 37, 95, 40, 62] for more details. 2.1 First definitions and properties Let (X; d) be a metric space. A geodesic segment in X is an isometrically embedded closed interval. A geodesic segment with endpoints x; y 2 X is denoted by [x; y]. Similarly are defined the geodesic rays and the bi-infinite geodesics in X. The space X is geodesic if every pair of points in X can be joined by a geodesic segment. Definition 2.1. A geodesic metric space X is hyperbolic if there is a constant δ ≥ 0 such that for every points x; y; z 2 X and every geodesic segments [x; y]; [y; z]; [z; x], one has [x; y] ⊂ Nδ([yz] [ [zx]). A companion definition is : Definition 2.2. 1. A map F : X1 ! X2 between metric spaces is a quasi-isometric embedding, if there are constants λ ≥ 1 and C ≥ 0 such that for every x; y 2 X1 λ−1d(x; y) − C ≤ d(F (x);F (y)) ≤ λd(x; y) + C: In the particular case where X1 is an interval in R, the map F is called a (λ, C)-quasi-geodesic of X2. 2. The map F is a quasi-isometry if it is a quasi-isometric embedding and if there is a constant R ≥ 0 such that NR(F (X1)) = X2. Examples are provided by the following situation. A group action is called geometric if it is an isometric, properly discontinuous and cocompact action. The Cayley graph of a finitely generated group (Γ;S), is the graph whose vertices are the elements of Γ, and whose edges are the pairs fg; gsg, with g 2 Γ and s 2 S [ S−1. It is endowed with the path metric obtained by identifying every edge with the unit interval. Proposition 2.3 (Svarc-Milnor). Suppose a group Γ acts geometrically on a geodesic metric space X. Let S be a finite generating set of Γ. Then for ev- ery O 2 X, the orbit map Γ ! X, g 7! g · O, extends to a quasi-isometry of the Cayley graph of (Γ;S) to X. 4 M. Bourdon The following result is called the Morse Lemma in [73]. Theorem 2.4. Suppose X is a δ-hyperbolic space. There is a constant R = R(λ, C; δ), such that for every (λ, C)-quasi-geodesic γ : I ! X there exists a geodesic c ⊂ X with γ(I) ⊂ NR(c). As a consequence one obtains Corollary 2.5. Suppose F : X1 ! X2 is a quasi-isometry between geodesic metric spaces. Then X1 is hyperbolic if and only if X2 is. Since the Cayley graphs of Γ, for the various choices of generators, are pairwise quasi-isometric, the following definition does not depend on the finite generating set S. Definition 2.6. A finitely generated group Γ is called hyperbolic if the Cayley graph of (Γ;S) is hyperbolic. 2.2 Boundary at infinity The boundary at infinity of a hyperbolic space, equipped with a visual metric, is the main protagonist of the paper. The boundary at infinity is defined in Definition 2.7, the visual metrics are defined in the statement of Theorem 2.9. Definition 2.7. Let X be a geodesic metric space. Geodesic rays r1; r2 : [0+1) ! X are asymptotic if sup d(r1(t); r2(t)) < 1: t2[0;+1) The boundary at infinity of X is @X := fr : [0 + 1) ! X geodesic rayg= ∼ where r1 ∼ r2 when they are asymptotic. It is endowed with the topology induced by the topology of uniform convergence on the compact subsets of [0; +1). The group Isom(X) acts on @X by homeomorphisms. Recall that a metric space is proper if its closed balls are compact. Using Ascoli's theorem one has : Proposition 2.8. Suppose that X is a proper hyperbolic space. 1. Given an origin O 2 X, every geodesic ray of X is asymptotic to a geodesic ray starting at O. 2. For every pair of non-asymptotic geodesic rays r1; r2, there is a geodesic γ : R ! X that is asymptotic to r1 for t ≤ 0 and to r2 for t ≥ 0. 3. The boundary @X is compact. In the sequel, we denote by (z; w) every geodesic which is asymptotic to a pair of distinct points z; w 2 @X.